Your TV display looks better than mine - I was hoping to use the mini to access the TV guide online but the quality of display in MacOS is dire. The system works perfectly for video and stored media in general, but not MacOS. Display quality on a different TV is rather better.
That said, for the benefit of the OP, the mini connects to almost any TV. Some TVs have VGA or even DVI inputs, in which case no adapters are needed, but others have Composite (yellow colored video and red and white audio) RCA connectors, or S-video, and for either of these, an adaptor is necessary such as the one linked to previously. In it's simplest form, audio can be nothing more complex than a stereo mini-jack to RCA cable to connect the mini's audio out to the TV's audio in, though you can also run audio to external stereo/home theatre systems the same way.